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Register a new userExtreme wave events for a nonlinear Schrodinger equation with linear damping and Gaussian driving
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We perform a numerical study of the initial-boundary value problem, with vanishing boundary conditions, of a driven nonlinear Schrodinger equation (NLS) with linear damping and a Gaussian driver. We identify Peregrine-like rogue waveforms, excited by two different types of vanishing initial data decaying at an algebraic or exponential rate. The observed extreme events emerge on top of a decaying support. Depending on the spatial/temporal scales of the driver, the transient dynamics -- prior to the eventual decay of the solutions -- may resemble the one in the semiclassical limit of the integrable NLS, or may, e.g., lead to large-amplitude breather-like patterns. The effects of the damping strength and driving amplitude, in suppressing or enhancing respectively the relevant features, as well as of the phase of the driver in the construction of a diverse array of spatiotemporal patterns, are numerically analyzed.
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We prove spatiotemporal algebraically decaying estimates for the density of the solutions of the linearly damped nonlinear Schrodinger equation with localized driving, when supplemented with vanishing boundary conditions. Their derivation is made via a scheme, which incorporates suitable weighted Sobolev spaces and a time-weighted energy method. Numerical simulations examining the dynamics (in the presence of physically relevant examples of driver types and driving amplitude/linear loss regimes), showcase that the suggested decaying rates, are proved relevant in describing the transient dynamics of the solutions, prior their decay: they support the emergence of waveforms possessing an algebraic space-time localization (reminiscent of the Peregrine soliton) as first events of the dynamics, but also effectively capture the space-time asymptotics of the numerical solutions.
We show that a nonlinear Schrodinger wave equation can reproduce all the features of linear quantum mechanics. This nonlinear wave equation is obtained by exploring, in a uniform language, the transition from fully classical theory governed by a nonlinear classical wave equation to quantum theory. The classical wave equation includes a nonlinear classicality enforcing potential which when eliminated transforms the wave equation into the linear Schrodinger equation. We show that it is not necessary to completely cancel this nonlinearity to recover the linear behavior of quantum mechanics. Scaling the classicality enforcing potential is sufficient to have quantum-like features appear and is equivalent to scaling Plancks constant.
We discuss the finite-time collapse, also referred as blow-up, of the solutions of a discrete nonlinear Schr{o}dinger (DNLS) equation incorporating linear and nonlinear gain and loss. This DNLS system appears in many inherently discrete physical contexts as a more realistic generalization of the Hamiltonian DNLS lattice. By using energy arguments in finite and infinite dimensional phase spaces (as guided by the boundary conditions imposed), we prove analytical upper and lower bounds for the collapse time, valid for both the defocusing and focusing cases of the model. In addition, the existence of a critical value in the linear loss parameter is underlined, separating finite time-collapse from energy decay. The numerical simulations, performed for a wide class of initial data, not only verified the validity of our bounds, but also revealed that the analytical bounds can be useful in identifying two distinct types of collapse dynamics, namely, extended or localized. Pending on the discreteness /amplitude regime, the system exhibits either type of collapse and the actual blow-up times approach, and in many cases are in excellent agreement, with the upper or the lower bound respectively. When these times lie between the analytical bounds, they are associated with a nontrivial mixing of the above major types of collapse dynamics, due to the corroboration of defocusing/focusing effects and energy gain/loss, in the presence of discreteness and nonlinearity.
We study the escape of a chain of coupled units over the barrier of a metastable potential. It is demonstrated that a very weak external driving field with suitably chosen frequency suffices to accomplish speedy escape. The latter requires the passage through a transition state the formation of which is triggered by permanent feeding of energy from a phonon background into humps of localised energy and elastic interaction of the arising breather solutions. In fact, cooperativity between the units of the chain entailing coordinated energy transfer is shown to be crucial for enhancing the rate of escape in an extremely effective and low-energy cost way where the effect of entropic localisation and breather coalescence conspire.
We discuss spatial dynamics and collapse scenarios of localized waves governed by the nonlinear Schr{o}dinger equation with nonlocal nonlinearity. Firstly, we prove that for arbitrary nonsingular attractive nonlocal nonlinear interaction in arbitrary dimension collapse does not occur. Then we study in detail the effect of singular nonlocal kernels in arbitrary dimension using both, Lyapunoffs method and virial identities. We find that for for a one-dimensional case, i.e. for $n=1$, collapse cannot happen for nonlocal nonlinearity. On the other hand, for spatial dimension $ngeq2$ and singular kernel $sim 1/r^alpha$, no collapse takes place if $alpha<2$, whereas collapse is possible if $alphage2$. Self-similar solutions allow us to find an expression for the critical distance (or time) at which collapse should occur in the particular case of $sim 1/r^2$ kernels. Moreover, different evolution scenarios for the three dimensional physically relevant case of Bose Einstein condensate are studied numerically for both, the ground state and a higher order toroidal state with and without an additional local repulsive nonlinear interaction. In particular, we show that presence of an additional local repulsive term can prevent collapse in those cases.
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